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Lesson plan Grade 5 Math

Perrindale Recharge Beds: A Grade 5 Fraction Area Lesson

A 45-minute grade 5 lesson finding the area of a fictional town's gravel recharge beds when the side lengths are fractions and mixed numbers of a yard.

Objectives

  1. Students will find the area of a rectangle whose side lengths are fractions of a yard.
  2. Students will multiply mixed numbers by rewriting each one as a fraction greater than one.
  3. Students will explain why multiplying by a fraction less than one gives a product smaller than the other factor.

Vocabulary

Aquifer
An aquifer is an underground layer of rock, sand, or gravel that holds water in the tiny spaces between its grains. Wells reach down into aquifers to bring water up to the surface.

Also: recharge bed, area, mixed number, square yard

Teacher overview

Perrindale is a fictional town that returns cleaned water to the ground instead of piping it to a river. Crews spread the water across shallow gravel beds, where it soaks down through sand and gravel and rejoins the groundwater below, a step that closes the local part of the water cycle. The beds are laid out in a tight space between a road and a hillside, so their sides come out in fractions and mixed numbers of a yard rather than in whole yards. The misconception this lesson corrects is multiplying mixed numbers piece by piece, so that 2 1/2 times 1 1/2 becomes 2 1/4 instead of 3 3/4, which happens when students multiply the whole numbers together and the fractions together and stop there.

Procedure

Hook (5 min): Hold up a sheet of grid paper and cut it so that it stands for a piece one half of a yard long and one half of a yard wide at the scale you are drawing with on the board. Ask students to estimate how much of a full square yard is left in your hands. Record two or three estimates, then say that the Perrindale crew has to answer this question for every bed it builds.

Direct instruction (12 min): Draw a unit square on the board and shade a rectangle two fifths across and one half up, showing that the shaded piece is two of the ten equal parts, or 1/5 of a square yard, which matches 2/5 x 1/2. Then move to a mixed-number bed, 3 1/2 yards by 1 1/2 yards. Rewrite each factor as a fraction greater than one, 7/2 and 3/2, multiply to get 21/4, and rename that as 5 1/4 square yards. Beside it, write the incorrect piece-by-piece answer of 3 1/4 and ask students to explain what that method leaves out.

Guided practice (10 min): Pairs find the area of a bed 2 1/4 yards long and 2 yards wide, rewriting 2 1/4 as 9/4 and finding 18/4, or 4 1/2 square yards. As a formative check, ask each pair to hold up a card with their answer. If more than a quarter of the class shows a number other than 4 1/2, work one more mixed-number problem together before the handout.

Independent practice (13 min): Students complete handout questions 1 through 6 alone. Circulate and watch for two errors: adding the side lengths instead of multiplying them, and multiplying mixed numbers piece by piece.

Closure and exit ticket (5 min): Students answer handout question 7 on an index card and turn it in on the way out. Sort the cards into correct and incorrect piles so you know who needs the area-model scaffold next time.

Perrindale Recharge Beds: A Grade 5 Fraction Area Lesson — Grade 5 Math
Name: ____________________________ Date: ______________

Student Handout

Perrindale spreads clean water on gravel beds so it can soak back into the ground. Each bed is a rectangle. Find the area of each bed in square yards. Write each answer in lowest terms.

  1. A test bed is 3/4 yard long and 1/2 yard wide. What is its area?
  2. A second test bed is 2/3 yard long and 3/5 yard wide. What is its area?
  3. A gravel strip is 4 yards long and 2/3 yard wide. What is its area?
  4. A bed is 2 1/2 yards long and 1 1/2 yards wide. What is its area?
  5. A bed is 3 1/2 yards long and 2 yards wide. What is its area?
  6. A bed is 6 yards long and 3/4 yard wide. Explain how you know its area is less than 6 square yards without finishing the multiplication.
  7. A bed is 1 1/4 yards long and 2/3 yard wide. What is its area?

From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.

Answer Key
  1. Answer: 3/8 square yard.
    Multiply the two side lengths: 3/4 x 1/2 = 3/8 square yard. An area model of a unit square cut into eighths shows three shaded parts.
  2. Answer: 2/5 square yard.
    Multiply to get 2/3 x 3/5 = 6/15, and 6/15 in lowest terms is 2/5 square yard.
  3. Answer: 2 2/3 square yards.
    Multiply 4 x 2/3 = 8/3, and 8/3 written as a mixed number is 2 2/3 square yards.
  4. Answer: 3 3/4 square yards.
    Rewrite the factors as 5/2 and 3/2. Then 5/2 x 3/2 = 15/4, and 15/4 written as a mixed number is 3 3/4 square yards.
  5. Answer: 7 square yards.
    Rewrite 3 1/2 as 7/2. Then 7/2 x 2 = 14/2, which is 7 square yards.
  6. Answer: The area is less than 6 square yards.
    The bed is only three fourths as wide as a bed 6 yards by 1 yard would be, and taking three fourths of a number gives a result smaller than that number, so the area must be under 6 square yards.
  7. Answer: 5/6 square yard.
    Rewrite 1 1/4 as 5/4. Then 5/4 x 2/3 = 10/12, and 10/12 in lowest terms is 5/6 square yard.

Differentiation

Support

Give these students grid paper already ruled into fourths in one direction and thirds in the other, so shading a rectangle and counting the small parts produces the product directly. Provide the rewritten fractions for questions 4 and 5, such as 5/2 and 3/2, and let students perform only the multiplication until renaming mixed numbers becomes steady.

Extension

Ask students to find two different pairs of side lengths, each with at least one fraction, that give a bed of exactly 4 1/2 square yards. Have them draw both beds to scale on grid paper and write which shape would fit better in a strip of land 3 yards wide.

ELL supports

Pre-teach "length," "width," "area," and "soak in" with a labeled drawing of a gravel bed and a hand pouring water into it, written in English and, where possible, in students' home languages. Provide a sentence frame for question 6: "The area is less than 6 square yards because 3/4 is less than ___."

Extensions

Have students measure a rectangular patch of bare ground or mulch on the school grounds to the nearest quarter yard, then find its area as a product of two mixed numbers. Ask them to write one sentence about whether rain landing on that patch would soak in or run off, and what the class could add to the patch to help it soak in.